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Norms and Basic Statistics

Norms and Basic Statistics. Cal State Northridge  427 Andrew Ainsworth PhD. Statistics AGAIN?. What do we want to do with statistics? Organize and Describe patterns in data Taking incomprehensible data and converting it to: Tables that summarize the data Graphs

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Norms and Basic Statistics

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  1. Norms and Basic Statistics Cal State Northridge 427 Andrew Ainsworth PhD

  2. Statistics AGAIN? • What do we want to do with statistics? • Organize and Describe patterns in data • Taking incomprehensible data and converting it to: • Tables that summarize the data • Graphs • Extract (i.e. INFER) meaning from data • Infer POPULATION values from SAMPLES • Hypothesis Testing – Groups • Hypothesis Testing – Relation/Prediction Psy 427 - Cal State Northridge

  3. Descriptives • Disorganized Data Psy 427 - Cal State Northridge

  4. Descriptives • Reducing and Describing Data Psy 427 - Cal State Northridge

  5. Descriptives • Displaying Data Psy 427 - Cal State Northridge

  6. Inferential • Inferential statistics: • Is a set of procedures to infer information about a population based upon characteristics from samples. • Samples are taken from Populations • Sample Statistics are used to infer population parameters Psy 427 - Cal State Northridge

  7. Inferential • Population is the complete set of people, animals, events or objects that share a common characteristic • A sample is some subset or subsets, selected from the population. • representative • simple random sample. Psy 427 - Cal State Northridge

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  9. Inferential • Does the number of hours students study per day affect the grade they are likely to receive in statistics (320)? Psy 427 - Cal State Northridge

  10. Inferential • Sometimes manipulation is not possible • Is prediction possible? • Can a relationship be established? • E.g., number of cigarettes smoked by per and the likelihood of getting lung cancer, • The level of child abuse in the home and the severity of later psychiatric problems. • Use of the death penalty and the level of crime. Psy 427 - Cal State Northridge

  11. -1 0 1 Inferential • Measured constructs can be assessed for co-relation (where the “coefficient of correlation” varies between -1 to +1) • “Regression analysis” can be used to assess whether a measured construct predicts the values on another measured construct (or multiple) (e.g., the level of crime given the level of death penalty usage). Psy 427 - Cal State Northridge

  12. Measurement • Statistical analyses depend upon the measurement characteristics of the data. • Measurement is a process of assigning numbers to constructs following a set of rules. • We normally measure variables into one of four different levels of measurement: • Nominal • Ordinal • Interval • Ratio Psy 427 - Cal State Northridge

  13. Contains 2 pieces of information D C B SIZE Ordinal Measurement • Where Numbers Representative Relative Size Only Psy 427 - Cal State Northridge

  14. D C B Numbers representing Size 1 2 3 Diff in numbers 2-1=1 3-2=1 Diff in size Size C – Size B =Size X Size D – Size C = Size X SIZE Interval Measurement: • Where Equal Differences Between Numbers Represent Equal Differences in Size Psy 427 - Cal State Northridge

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  17. Measurement • Ratio Scale Measurement • In ratio scale measurement there are four kinds of information conveyed by the numbers assigned to represent a variable: • Everything Interval Measurement Contains Plus • A meaningful 0-point and therefore meaningful ratios among measurements. Psy 427 - Cal State Northridge

  18. True Zero point Psy 427 - Cal State Northridge

  19. Measurement • Ratio Scale Measurement • If we have a true ratio scale, where 0 represents an a complete absence of the variable in question, then we form a meaningful ratio among the scale values such as: • However, if 0 is not a true absence of the variable, then the ratio 4/2 = 2 is not meaningful. Psy 427 - Cal State Northridge

  20. Percentiles and Percentile Ranks • A percentile is the score at which a specified percentage of scores in a distribution fall below • To say a score 53 is in the 75th percentile is to say that 75% of all scores are less than 53 • The percentile rank of a score indicates the percentage of scores in the distribution that fall at or below that score. • Thus, for example, to say that the percentile rank of 53 is 75, is to say that 75% of the scores on the exam are less than 53. Psy 427 - Cal State Northridge

  21. Percentile • Scores which divide distributions into specific proportions • Percentiles = hundredths P1, P2, P3, … P97, P98, P99 • Quartiles = quarters Q1, Q2, Q3 • Deciles = tenths D1, D2, D3, D4, D5, D6, D7, D8, D9 • Percentiles are the SCORES Psy 427 - Cal State Northridge

  22. Percentile Rank • What percent of the scores fall below a particular score? • Percentile Ranks are the Ranks not the scores Psy 427 - Cal State Northridge

  23. Example: Percentile Rank • Ranking no ties – just number them • Ranking with ties - assign midpoint to ties Psy 427 - Cal State Northridge

  24. Steps to Calculating Percentile Ranks • Example: Psy 427 - Cal State Northridge

  25. Percentile Where XP is the score at the desired percentile, p is the desired percentile (a number between 0 and 1) and n is the number of scores) If the number is an integer, than the desired percentile is that number If the number is not an integer than you can either round or interpolate; for this class we’ll just round (round up when p is below .50 and down when p is above .50) Psy 427 - Cal State Northridge

  26. Percentile • Apply the formula • You’ll get a number like 7.5 (think of it as place1.proportion) • Start with the value indicated by place1 (e.g. 7.5, start with the value in the 7th place) • Find place2 which is the next highest placenumber (e.g. the 8th place) and subtract the value in place1 from the value in place2, this distance1 • Multiple the proportion number by the distance1 value, this is distance2 • Add distance2 to the value in place1 and that is the interpolated value Psy 427 - Cal State Northridge

  27. Example: Percentile • Example 1: 25th percentile: {1, 4, 9, 16, 25, 36, 49, 64, 81} • X25 = (.25)(9+1) = 2.5 • place1 = 2, proportion = .5 • Value in place1 = 4 • Value in place2 = 9 • distance1 = 9 – 4 = 5 • distance2 = 5 * .5 = 2.5 • Interpolated value = 4 + 2.5 = 6.5 • 6.5 is the 25th percentile Psy 427 - Cal State Northridge

  28. Example: Percentile • Example 2: 75th percentile {1, 4, 9, 16, 25, 36, 49, 64, 81} • X75 = (.75)(9+1) = 7.5 • place1 = 7, proportion = .5 • Value in place1 = 49 • Value in place2 = 64 • distance1 = 64 – 49 = 15 • distance2 = 15 * .5 = 7.5 • Interpolated value = 49 + 7.5 = 56.5 • 56.5 is the 75th percentile Psy 427 - Cal State Northridge

  29. Quartiles • To calculate Quartiles you simply find the scores the correspond to the 25, 50 and 75 percentiles. • Q1 = P25, Q2 = P50, Q3 = P75 Psy 427 - Cal State Northridge

  30. Reducing Distributions • Regardless of numbers of scores, distributions can be described with three pieces of info: • Central Tendency • Variability • Shape (Normal, Skewed, etc.) Psy 427 - Cal State Northridge

  31. Measures of Central Tendency Psy 427 - Cal State Northridge

  32. The Mean • Only used for interval & ratio data. • Major advantages: • The sample value is a very good estimate of the population value. Psy 427 - Cal State Northridge

  33. Reducing Distributions • Regardless of numbers of scores, distributions can be described with three pieces of info: • Central Tendency • Variability • Shape (Normal, Skewed, etc.) Psy 427 - Cal State Northridge

  34. How do scores spread out? • Variability • Tell us how far scores spread out • Tells us how the degree to which scores deviate from the central tendency Psy 427 - Cal State Northridge

  35. Mean = 10 Mean = 10 How are these different? Psy 427 - Cal State Northridge

  36. Measure of Variability Psy 427 - Cal State Northridge

  37. The Range • The simplest measure of variability • Range (R) = Xhighest – Xlowest • Advantage – Easy to Calculate • Disadvantages • Like Median, only dependent on two scores  unstable {0, 8, 9, 9, 11, 53} Range = 53 {0, 8, 9, 9, 11, 11} Range = 11 • Does not reflect all scores Psy 427 - Cal State Northridge

  38. Variability: IQR • Interquartile Range • = P75 – P25 or Q3 – Q1 • This helps to get a range that is not influenced by the extreme high and low scores • Where the range is the spread across 100% of the scores, the IQR is the spread across the middle 50% Psy 427 - Cal State Northridge

  39. Variability: SIQR • Semi-interquartile range • =(P75 – P25)/2 or (Q3 – Q1)/2 • IQR/2 • This is the spread of the middle 25% of the data • The average distance of Q1 and Q3 from the median • Better for skewed data Psy 427 - Cal State Northridge

  40. Variability: SIQR • Semi-Interquartile range Q1 Q2 Q3 Q1 Q2 Q3 Psy 427 - Cal State Northridge

  41. Variance • The average squareddistance of each score from the mean • Also known as the mean square • Variance of a sample: s2 • Variance of a population: s2 Psy 427 - Cal State Northridge

  42. When calculated for a sample When calculated for the entire population Variance Psy 427 - Cal State Northridge

  43. Standard Deviation • Variance is in squared units • What about regular old units • Standard Deviation = Square root of the variance Psy 427 - Cal State Northridge

  44. Standard Deviation • Uses measure of central tendency (i.e. mean) • Uses all data points • Has a special relationship with the normal curve • Can be used in further calculations • Standard Deviation of Sample = SD or s • Standard Deviation of Population =  Psy 427 - Cal State Northridge

  45. Why N-1? • When using a sample (which we always do) we want a statistic that is the best estimate of the parameter Psy 427 - Cal State Northridge

  46. Degrees of Freedom • Usually referred to as df • Number of observations minus the number of restrictions __+__+__+__=10 - 4 free spaces 2 +__+__+__=10 - 3 free spaces 2 + 4 +__+__=10 - 2 free spaces 2 + 4 + 3 +__=10 Last space is not free!! Only 3 dfs. Psy 427 - Cal State Northridge

  47. Reducing Distributions • Regardless of numbers of scores, distributions can be described with three pieces of info: • Central Tendency • Variability • Shape (Normal, Skewed, etc.) Psy 427 - Cal State Northridge

  48. Psy 427 - Cal State Northridge

  49. Psy 427 - Cal State Northridge

  50. Example: The Mean = 100 and the Standard Deviation = 20 Psy 427 - Cal State Northridge

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